Books

• Jean Claude, Morphometry with R
  • Online through ETHZ
  • Buy it
• John C. Russ, “The Image Processing Handbook”,(Boca Raton, CRC Press)
  • Available online within domain ethz.ch (or proxy.ethz.ch / public VPN)
• J. Weickert, Visualization and Processing of Tensor Fields
  • Online

Papers / Sites

• Voronoi Tesselations

  • Ghosh, S. (1997). Tessellation-based computational methods for the characterization and analysis of heterogeneous microstructures. Composites Science and Technology, 57(9-10), 1187–1210
  • Wolfram Explanation

• Self-Avoiding / Nearest Neighbor

  • Schwarz, H., & Exner, H. E. (1983). The characterization of the arrangement of feature centroids in planes and volumes. Journal of Microscopy, 129(2), 155–169.
  • Kubitscheck, U. et al. (1996). Single nuclear pores visualized by confocal microscopy and image processing. Biophysical Journal, 70(5), 2067–77.

• Alignment / Distribution Tensor

  • Mader, K. et al (2013). A quantitative framework for the 3D characterization of the osteocyte lacunar system. Bone, 57(1), 142–154
  • Aubouy, M., et al. (2003). A texture tensor to quantify deformations. Granular Matter, 5, 67–70. Retrieved from http://arxiv.org/abs/cond-mat/0301018

• Two point correlation

  • Dinis, L., et. al. (2007). Analysis of 3D solids using the natural neighbour radial point interpolation method. Computer Methods in Applied Mechanics and Engineering, 196(13-16)

• Motivation (Why and How?)
• Local Environment
  • Neighbors
  • Voronoi Tesselation
  • Distribution Tensor

• Global Enviroment
  • Alignment
  • Self-Avoidance
  • Two Point Correlation Function

We examine a number of different metrics in this lecture and additionally to classifying them as Local and Global we can define them as point and voxel-based operations.

Point Operations

• Nearest Neighbor
• Delaunay Triangulation
  • Distribution Tensor
• Point (Center of Volume)-based Voronoi Tesselation
• Alignment

Voxel Operation

• Voronoi Tesselation
• Neighbor Counting
• 2-point (N-point) correlation functions

With all of these images, the first step is always to understand exactly what we are trying to learn from our images.

1. We want to know how many cells are alive

   • Maybe small cells are dead and larger cells are alive \( \rightarrow \) examine the volume distribution
   • Maybe living cells are round and dead cells are really spiky and pointy \( \rightarrow \) examine anisotropy

2. We want to know where the cells are alive or most densely packed

   • We can visually inspect the sample (maybe even color by volume)
   • We can examine the raw positions (x,y,z) but what does that really tell us?
   • We can make boxes and count the cells inside each one
   • How do we compare two regions in the same sample or even two samples?

1. We want to know how the cells are communicating
   • Maybe physically connected cells (touching) are communicating \( \rightarrow \) watershed
   • Maybe cells oriented the same direction are communicating \( \rightarrow \) average? orientation
   • Maybe cells which are close enough are communicating \( \rightarrow \) ?
   • Maybe cells form hub and spoke networks \( \rightarrow \) ?

1. We want to know how the cells are nourished
   • Maybe closely packed cells are better nourished \( \rightarrow \) count cells in a box?
   • Maybe cells are oriented around canals which supply them \( \rightarrow \) ?

1. A way for counting cells in a region and estimating density without creating arbitrary boxes
2. A way for finding out how many cells are near a given cell, it's nearest neighbors
3. A way for quantifying how far apart cells are and then comparing different regions within a sample
4. A way for quantifying and comparing orientations

What would be really great?

A tool which could be adapted to answering a large variety of problems

• multiple types of structures
• multiple phases

With most imaging techniques and sample types, the task of measurement itself impacts the sample.

• Even techniques like X-ray tomography which claim to be non-destructive still impart significant to lethal doses of X-ray radition for high resolution imaging
• Electron microscopy, auto-tome-based methods, histology are all markedly more destructive and make longitudinal studies impossible
• Even when such measurements are possible
  • Registration can be a difficult task and introduce artifacts

Why is this important?

• techniques which allow us to compare different samples of the same type.
• are sensitive to common transformations
  • Sample B after the treatment looks like Sample A stretched to be 2x larger
  • The volume fraction at the center is higher than the edges but organization remains the same

\[ \downarrow \]

\[ \cdots \]

So if we want to know the the mean or standard deviations of the position or orientations we can analyze them easily.

• But what if we want more or other information?

When given a group of data, it is common to take a mean value since this is easy. The mean bone thickness is 0.3mm. This is particularly relevant for groups with many samples because the mean is much smaller than all of the individual points.

but means can lie

• the mean of 0\( ^\circ \) and 180\( ^\circ \) = 90\( ^\circ \)
• the distance between -180\( ^\circ \) and 179\( ^\circ \) is 359\( ^\circ \)
• since we have not defined a tip or head, 0\( ^\circ \) and 180\( ^\circ \) are actually the same

some means are not very useful

One of the first metrics to examine with distribution is density \( \rightarrow \) how many objects in a given region or volume.

It is deceptively easy to calculate involving the ratio of the number of objects divided by the volume.

It doesn't tell us much, many very different systems with the same density and what if we want the density of a single point? Does that even make sense?

Definition

Oxford American \( \rightarrow \) be situated next to or very near to (another)

• Does not sound very scientific
• How close?
  • Touching, closer than anything else?

Nearest Neighbor (distance)

Given a set of objects with centroids at \[ \textbf{P}=\begin{bmatrix} \vec{x}_0,\vec{x}_1,\cdots,\vec{x}_i \end{bmatrix} \]

We can define the nearest neighbor as the position of the object in our set which is closest

\[ \vec{\textrm{NN}}(\vec{y}) = \textrm{argmin}(||\vec{y}-\vec{x}|| \forall \vec{x} \in \textbf{P}-\vec{y}) \]

We define the distance as the Euclidean distance from the current point to that point, and the angle as the

\[ \textrm{NND}(\vec{y}) = \textrm{min}(||\vec{y}-\vec{x}|| \forall \vec{x} \in \textbf{P}-\vec{y}) \] \[ \textrm{NN}\theta(\vec{y}) = \tan^{-1}\frac{(\vec{\textrm{NN}}-\vec{y})\cdot \vec{j}}{(\vec{\textrm{NN}}-\vec{y})\cdot \vec{i}} \]

So examining a simple starting system like a grid, we already start running into issues.

• In a perfect grid like structure each object has 4 equidistant neighbors (6 in 3D)
• Which one is closest?

We thus add an additional clause (only relevant for simulated data) where if there are multiple equidistant neighbors, a nearest is chosen randomly

This ensures when we examine the orientation distribution (NN\( \theta \)) of the neighbors it is evenly distributed

For the rest of these sections we will repeatedly use several simple in-silico systems to test our methods and try to better understand the kind of results we obtain from them.

• Compression

  • The most simple system simply involves a scaling in every direction by \( \alpha \)
  • \( \alpha<1 \) the system is compressed
  • \( \alpha>1 \) the system is expanded

  \[ \begin{bmatrix} x^\prime \\ y^\prime \end{bmatrix} = \alpha \begin{bmatrix} x \\ y \end{bmatrix} \]

• Stretch
  • A non-evenly distributed system with a parameter \( \alpha \) controlling if objects bunch near the edges or the center. A maximum distance \( m \) is defined as the magnitude of the largest offset in x or y
  • + Same total volume just arranged differently

\[ \begin{bmatrix} x^\prime \\ y^\prime \end{bmatrix} = \begin{bmatrix} \textrm{sign}(x) \left(\frac{|x|}{m}\right)^\alpha m \\ \textrm{sign}(y) \left(\frac{|y|}{m}\right)^\alpha m \end{bmatrix} \]

• Swirl
  • A transformation where the points are rotated more based on how far away they are from the center and the slope of the swirl (\( \alpha \)),

\[ \theta (x,y) = \alpha \sqrt{x^2+y^2} \]

\[ \begin{bmatrix} x^\prime \\ y^\prime \end{bmatrix} = \begin{bmatrix} \cos\theta(x,y) & -\sin\theta(x,y) \\ \sin\theta(x,y) & \cos\theta(x,y) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]

Using a uniform grid of points as a starting point has a strong influence on the results. A better approach is to use a randomly distributed series of points

• resembles real data much better
• avoids these symmetry problems
  • \( \epsilon \) sized edges or overlaps
  • identical distances to nearby objects